Problem 29 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (3,-4),(3,16) $$

Step-by-Step Solution

Answer

The slope is undefined (a vertical line).

Step 1: Understand the Problem

To find the slope of a line passing through two points, we use the slope formula. The points given are \((x_1, y_1) = (3, -4)\) and \((x_2, y_2) = (3, 16)\).

Step 2: Review the Slope Formula

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Substitute the Values into the Formula

Plug the given points into the slope formula: \[ m = \frac{16 - (-4)}{3 - 3} \] Simplifying the numerator gives: \(16 - (-4) = 16 + 4 = 20\).The denominator is \(3 - 3 = 0\).

Step 4: Analyze the Result

Since the denominator is zero, the slope is undefined. This indicates that the line is vertical.

Key Concepts

Understanding Undefined Slope

When discussing the concept of an undefined slope, it's essential to understand what slope represents. The slope of a line is a measure of its steepness and direction. We calculate it as the change in the vertical direction (rise) over the change in the horizontal direction (run).

But what happens when the run is zero? If you attempt to divide any number by zero, you get an undefined result. This scenario occurs when the slope is undefined.

• A line with an undefined slope runs vertically.
• This means it has no corresponding horizontal change, making the denominator in the slope formula zero.

If the points of a line share the same x-coordinate, like in our example, the slope becomes undefined.

The Nature of Vertical Lines

In our exercise, you might have observed that both points, rr(3, -4) and (3, 16), have an identical x-coordinate. Vertically aligned points on a graph indicate a vertical line.

Vertical lines are unique because they contain points with the same x-values but different y-values, which makes calculating their slopes with the usual formula impossible.

• Vertical lines are represented by the equation: \( x = a \), where \( a \) is the constant x-coordinate value for every point on the line.
• They extend indefinitely in the vertical direction.
• This unique characteristic renders their slope undefined.

In simple terms, their direction is purely up and down, hence no horizontal transition occurs on these lines.

Applying the Slope Formula

The slope formula is a fundamental tool when working with linear equations and graphing. To find the slope, use: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

The formula tells us to calculate the difference in y-values and divide it by the difference in x-values. This approach works smoothly unless the line is vertical.

• For our example, the points \((x_1, y_1) = (3, -4)\) and \((x_2, y_2) = (3, 16)\) yield a denominator of zero.
• Calculating the numerator by subtracting the y-values gives 20.

But since we have zero in the denominator, the slope is undefined. Remember: an undefined slope is always paired with a vertical line.